Rotational friction on small globular proteins: Combined
dielectric and hydrodynamic effect
Arnab Mukherjee and Biman Bagchi
∗
arXiv:physics/0408071 v1 15 Aug 2004
Solid State and Structural Chemistry Unit, Indian Institute of
Science, Bangalore, India 560 012.
Abstract
Rotational friction on proteins and macromolecules is known to derive contributions from at least two distinct sources – hydrodynamic
(due to viscosity) and dielectric friction (due to polar interactions).
In the existing theoretical approaches, the effect of the latter is taken
into account in an ad hoc manner, by increasing the size of the protein with the addition of a hydration layer. Here we calculate the
rotational dielectric friction on a protein (ζDF ) by using a generalized
arbitrary charge distribution model (where the charges are obtained
from quantum chemical calculation) and the hydrodynamic friction
stick
with stick boundary condition, (ζhyd
) by using the sophisticated theoretical technique known as tri-axial ellipsoidal method, formulated
by Harding [S. E. Harding, Comp. Biol. Med. 12, 75 (1982)]. The
calculation of hydrodynamic friction is done with only the dry volume
of the protein (no hydration layer). We find that the total friction
stick
obtained by summing up ζDF and ζhyd
gives reasonable agreement
stick
with the experimental results, i.e., ζexp ≈ ζDF + ζhyd
.
∗
Email: bbagchi@sscu.iisc.ernet.in
1
1
Introduction
In this article, we present an interesting result that the experimentally
observed rotational correlation time of a large number of proteins can
essentially be described as the combined effect of the rotational dielectric and hydrodynamic frictions on the proteins. Thus, one needs
not assume the existence of a rigid hydration layer around the protein, as is often assumed in the standard theoretical calculations of
hydrodynamic friction.
The study of rotational friction of proteins in aqueous solution has a
long history [1−12]. Despite many decades of study, several aspects of
the problem remain ill understood. For proteins and macromolecules,
the rotational friction is obtained from Debye-Stokes-Einstein (DSE)
relation given by,
ζR = 8πη R3 ,
(1)
where ζR is the rotational friction on the protein and R is the radius of
the protein. Naturally, the above relation assumes a spherical shape
of the protein, which is often not correct. Moreover, there is ambiguity about the determination of some average radius of the protein. If
one obtains the radius from the standard mass density of the protein
(0.73 gm/cc), the values of the rotational friction are much smaller.
The dielectric measurement of Grant [4] showed that the experimental
value of rotational friction of myoglobin could only be explained by the
above DSE equation, if one assumes a thick hydration layer around
the protein, thereby increasing the radius of the protein. It is well
known that spherical approximation embedded in DSE is grossly in
error and the shape of the protein is quite important. However, even
with the more recent sophisticated techniques such as tri-axial ellipsoid
method [5] and the microscopic bead modeling technique [6, 7], which
take due recognition of the non-spherical shape of the macromolecule,
agreement with the experimental result is not possible without the
incorporation of a rigid hydration layer [10]. In should be recognized
that the effect of hydration layer thus introduced is purely ad hoc.
In the case of tri-axial ellipsoidal method, the values of the axes are
2
increased proportionately by increasing the percentage of encapsulation of the protein atoms inside its equivalent ellipsoid [11, 12]. On
the other hand, the microscopic bead modeling technique uses beads
of much bigger size [6] (3.0 Å instead of 1.2 Å) to take care of the
effect of hydration layer. Without the hydration layer, the estimate of
friction obtained from the theory is systematically lower.
It has been recognized quite early that water in the hydration layer
surrounding proteins and macromolecules has completely different dynamical properties than those in the bulk [13]. The dynamics of water molecules in the hydration layer are also subject of great interest
as they could play crucial role in the property and activity of these
molecules. One often discusses the crossover from biological activity
to the observed inactivity at low temperatures in terms of a proteinglass transition observed in the hydrated proteins [14]. Recent investigations have shown that the water molecules in the hydration layer
are not only more structured but they also show slow translational
and rotational motion than their bulk counterpart [15, 16, 17, 18, 19].
Nevertheless, it is highly unlikely that water molecules in the surface of a protein such as myoglobin are so slow that we can replace it
by a rigid hydration layer. On the contrary, all the recent experimental and simulation studies have shown that the water in the surface
of the protein exhibits bimodal dynamics [20]. Majority of the water molecules seem to retain their bulk-like dynamics while a fraction
(∼ 20%) exhibits markedly slow dynamics. Recent solvation dynamics and photon echo peak shift experiment not only established the
existence of slow water on the surface of proteins but also showed
that the hydration layer is quite labile [21]. If one defines an average
residence time to characterize the dynamics of water in the hydration
layer, the residence time of bound or quasi-bound water is expected to
range from 20 to 300 ps [22]. Question naturally arises how to understand quantitatively the role of the hydration layer in enhancing the
rotational friction on the protein molecules. Clearly, the picturesque
description of an immobile rigid layer around protein needs to be replaced by a description where the hydration layer is slow but definitely
3
dynamic.
This labile hydration layer has been explained in terms of a dynamic
exchange model, which assumes that due to the presence of relatively
stronger hydrogen bonding of water molecules with the charged groups
at the surface of the protein, a surface water molecule can exist in
either of the following two states – bound and free [23]. The free
water molecules have dynamical characteristics similar to those of the
bulk but the bound water molecules are essentially made static by the
hydrogen bonding with the surface. In this picture, the slow time scale
arises due to the dynamical exchange between the two states of the
water molecules. Recent computer simulations seem to have confirmed
the essential aspects of the dynamic exchange model (DEM) [24].
While the above model can provide a simple explanation of the
origin of the observed slow dynamics, its correlation with dynamical
properties of protein has not yet been established. This is a non-trivial
problem as discussed below.
The mode coupling theory (MCT) is another viable quantitative
theory, which has been quite successful in describing translational and
rational motion of small molecules [25]. This approach has also been
extended to treat dynamics of polymer and biomolecules [26]. Let us
recall a few of the lessons learned from the MCT of rotational friction
of small molecules, and translational friction of ions in dipolar liquids.
In both the cases, intermolecular dipole-dipole/ion-dipole correlations
were found to play important role. It was also found that if one
neglects the translational mode of the solvent molecules, then the friction on polar solute increases by several factors. It should be noted
here that the continuum models/hydrodynamic description of rotational friction always ignored this translational component. In fact,
this translational component plays a hidden role in reducing the effect
of the role of molecular level solute solvent and solvent-solvent pair
(both isotropic and orientational) correlations that increase the value
of the friction over the continuum model prediction. Thus, the issue
is rather involved. In fact, the continuum model is found to give ac4
curate results due to cancellation of two errors: neglect of short-range
correlations and neglect of translational contribution. In view of the
above, it is thus important to note that the slow water molecules in
the hydration layer can enhance the friction considerably. Thus, the
classical picture of rigid, static hydration layer needs to be replaced by
dynamic layer where the translational motion of the water molecules
should be related to the residence time. However, only preliminary
progress has been made in this direction. Thus, continuum models
remain the only theoretical method to treat dielectric friction on complex molecules.
An important issue in the calculation of the rotational friction is
that proteins are characterized by complex charge distribution. The
earliest models to estimate the enhanced friction on a probe, due
to the interactions of its polar groups with the surrounding water
molecules in an aqueous solution, employed a point dipole approximation [27, 28, 29]. In the simplest version of the model, the probe
molecule is replaced by a sphere with a point dipole at the center of the
sphere. Such an approach is reasonable for small molecules, although
continuum model itself may have certain limitations. The situation
is quite different for large molecules like proteins because the charge
here is distributed over a large volume and the surface charges are
close to the water molecules. Thus, the point dipole approximation
becomes inapplicable to such systems. This limitation of the early continuum models was removed by Alavi and Waldeck [30] who obtained
an elegant expression for the dielectric friction on a molecule with extended arbitrary charge distribution. By studying several well-known
dye molecules, they demonstrated that the extended charge distribution indeed has a strong effect on the dielectric friction on the probe
molecules. The work of Alavi and Waldeck [30] constitutes an important advance in the study of dielectric friction. The role of dielectric
friction has been studied for the organic molecules by other authors
[31].
The objective of the present work is to attempt to replace the rigid
hydration layer used in hydrodynamic calculation. To this goal, we
5
calculate the hydrodynamic friction using the tri-axial method [5], in
which the shape of a protein is mapped to an ellipsoid of three unequal axes – closely representing the shape and size of the protein. No
hydration layer is added in the calculation. We then calculate the dielectric friction using Alavi and Waldeck’s model of generalized charge
distribution for a large number of proteins. The friction contributions
obtained from the above two methods are combined to obtain the total
rotational friction. When compared, the total friction has been found
to agree closely with the experimental result.
We have also extended the work of Alavi and Waldeck to include
multiple shells of water with different dielectric constants around a
protein. The multiple shell model is introduced in concern with the
experimental observation of varying dielectric constants of water from
the hydration layer surrounding a protein to the bulk water. These
shells have distinct dielectric properties – both static and dynamic.
The resulting analytical expressions can be used to obtain quantitative
prediction of the effects of a slow layer of water molecules on the
dielectric friction on proteins. However, the multiple shell model in
the continuum fails since it adds up the friction in every layer. This
has been discussed in the appendix.
2
Results and Discussion
Below, we discuss the results obtained from the different aspects of
rotational friction of proteins. The coordinates of the proteins are
obtained from protein data bank (PDB) [32].
2.1
Dielectric Friction
Dielectric friction is an important part of rotational friction for polar
or charged molecules in polar solvent, because of the polarization of the
6
solvent medium. The solvent molecules, being polarized by the probe,
create a reaction filed, which opposes the rotation of the probe.
Many of the amino acid residues, which constitute the protein, are
polar or hydrophilic. Therefore, in the aqueous solution, a protein and
other polar molecules experience significant dielectric friction. There
exist several theories [27, 28, 29, 33, 34, 35], which account for the
dielectric contribution to the friction. Some of these theories are continuum model calculation of a point charge or point dipole rotating
within the spherical cavity. Nee and Zwanzig [27] provide an estimate
of dielectric friction on a point dipole in terms of the dipole moment
of the point dipole, dielectric constant of the solvent, Debye relaxation time, and the chosen cavity radius. Later, Alavi and Waldeck
[30] extended this theory to incorporate the arbitrary multiple charge
distribution of the probe molecule.
The dielectric friction on the proteins has been calculated from the
expression of Alavi and Waldeck for arbitrary multiple charge distribution model given below [30],
ζDF
l 2l + 1 (l − m)!
∞ X
N X
N X
X
ri l rj l
8 ǫs − 1
×
τD
qi qj
=
Rc (2ǫ1 + 1)2 j=1 i=1 l=1 m=1 l + 1 (l + m)!
Rc Rc
m2 Plm (cos θi ) Plm (cos θj )cos(mφji )
where Rc is the cavity radius, (ri, θi, φi) is the position vector and
qi is the partial charge of the ith atom. Plm (cos(θi) is the Legendre
polynomial. The maximum value of l used in the Legendre polynomial
is 50. ǫs is the static dielectric constant of the solvent. Since the
solvent here is water, ǫs is taken to be 78 and the Debye relaxation
time τD is taken as 8.3 picosecond (ps).
The partial charges (qi) of the atoms constituting the proteins have
been calculated using the extended Huckel model of the semi empirical
calculation package of Hyperchem software. The dielectric friction is
calculated on each of the atoms in a protein. The rotational frictions
around X, Y and Z direction are calculated by changing the labels
av
is the
of the atom coordinates. The average dielectric constant ζDF
7
(2)
harmonic mean of the dielectric frictions along X, Y and Z direction.
Here X, Y, and Z denote the space fixed Cartesian coordinate of the
proteins, as obtained from PDB [32].
Table 1 shows the values of dielectric friction along X, Y, Z direction
and their average. Continuum calculation method of the dielectric
friction formulated by Alavi and Waldeck is dependent on the cavity
radius and has been discussed in detail by them [30]. They calculated
the cavity radius from the observed orientational relaxation time of
the organic molecules. The ratios of the longest bond vector of the
organic molecules to the cavity radius ranged from 0.75 to 0.85. In
Table 1, the calculation s of dielectric friction are performed using the
cavity radius such that the ratio of the longest bond vector to the
cavity radius is 0.75.
In Table 2, we compared the average dielectric friction for the two
0.85
0.85
0.75
). ζDF
) and 0.85 (denoted as ζDF
above ratios – 0.75 (denoted as ζDF
0.75
is always larger than ζDF
since the shorter cavity radius will put the
charges close to the surface of the cavity, thereby increasing the polarization of the solvent and hence the rotational friction of the molecule.
2.2
Hydrodynamic Friction
The hydrodynamic rotational friction of the protein depends on its
shape and size. Hydrodynamic friction was estimated earlier by the
well-known DSE relation (Eq. 1). Perrin in 1936 [36] extended the
DSE theory to calculate the hydrodynamic friction for molecules with
prolate and oblate like shapes. Both prolate and oblate have two
unequal axes. Harding [5] further extended the theory to calculate
the hydrodynamic friction using a tri-axial ellipsoid. All the above
theories employ stick binary condition to obtain the hydrodynamic
friction.
Tri-axial ellipsoidal technique requires the construction of an equiv8
alent ellipsoid of the protein. We have followed the method of Taylor
et al. to construct an equivalent ellipsoid from the moment matrix
[37]. The eigenvalues of this equivalent ellipsoid are proportional to
the square of the axes. So this method provides with the two axial
ratios. We then obtained the values of the axes using the formula
given by Mittelbach [38]
1
(3)
Rγ2 = (A2 + B 2 + C 3)
5
Rγ is the radius of gyration and A, B and C are the three unequal
axes of a particular protein.
Once the protein is represented as an ellipsoid with three principle
axes, the hydrodynamic friction is calculated using Harding’s method
[5, 39]. The hydrodynamic rotational friction of the ellipsoidal axes A,
B and C are denoted as ζA , ζB and ζC . The above rotational friction
is obtained from the series of equations given below [39],
ζ0 = 8πηABC,
2(B 2 + C 2)
ζA = ζ0
,
3ABC(B 2α2 + C 2α3 )
2(A2 + C 2)
,
ζB = ζ0
3ABC(A2α1 + C 2α3 )
2(A2 + B 2)
ζC = ζ0
,
3ABC(A2α1 + B 2 α2 )
Z ∞
dλ
,
α1 =
0 (A2 + λ)∆
Z ∞
dλ
,
α2 =
0 (B 2 + λ)∆
Z ∞
dλ
,
α1 =
0 (C 2 + λ)∆
2
2
2
∆ = (A + λ)(B + λ)(C + λ)
9
1
2
(4)
η is the viscosity of the solvent. We have calculated the average of triaxial hydrodynamic friction by taking a harmonic mean of the friction
along three different axes, as given below,
1
ζTavR
1 1
1
1 =
+
+
3 ζTAR ζTBR ζTCR
(5)
The values of hydrodynamic friction, along three principle axes (A,
B and C) of the ellipsoid and their mean, are tabulated in the Table 3.
The A, B and C axes are not the same as the space fixed X, Y, Z Cartesian reference frame. Note that the values obtained from the tri-axial
method are much lower than the experimental values. Here, we can
talk about an important aspect of standard hydrodynamic approach
– hydration layer. One finds that hydrodynamic values of rotational
friction underestimate the rotational friction unless the effect of hydration layer is taken into account. However, the effect of hydration
layer is usually incorporated in an ad hoc manner, by increasing the
percentage of encapsulation of the atoms inside the ellipsoid [12, 11].
In this method, once the two axial ratios are obtained from the equivalent ellipsoid, the actual values of the axes are obtained by increasing
the encapsulation of the protein atoms inside the ellipsoid. In the calculation presented here, the axes are obtained by equating with the
radius of gyration. Therefore, we considered no hydration layer in
this calculation of hydrodynamic friction. Later, we will show that
this effect of hydration layer comes from the dielectric friction.
2.3
Total rotational friction: Comparison with experimental results
We define the total rotational friction as the sum of dielectric friction
av
(ζDF
) and the hydrodynamic friction without the hydration layer (i.e.
tri-axial friction, ζTavR ) as given below,
av
ζtotal = ζDF
+ ζTavR
10
(6)
av
In Table 4, we have shown the values of the average dielectric (ζDF
),
av
hydrodynamic (ζT R ) friction. Total friction (ζtotal ) defined above is
shown in the fourth column. To compare with the experimental results, we have shown the experimental values of the rotational friction
in the next column. Note here, while the total friction, which is the
contribution from both dielectric and hydrodynamic friction, is close
to the experimental result, the microscopic bead modeling predicts
the result, which is close to experimental value by itself [7]. The last
column of Table 4 shows the references of the articles from which the
experimental results are obtained.
The similarity between the total friction and the experimental friction is shown in figure 1, where we have plotted the experimental values of rotational friction against the total friction for a large number
of proteins. For most of the proteins, the results fall on the diagonal
line.
From the results shown in Table 4, we can conclude that the sum
of dielectric friction and the hydrodynamic friction of the dry protein
is approximately equal to the experimental results.
ζtotal ≈ ζexp
3
(7)
Conclusion
Let us first summarize the main results of this work. We have calculated the hydrodynamic rotational friction on proteins using the
tri-axial ellipsoid method, formulated by Harding [5], and the dielectric friction using the generalized charge distribution model derived
by Alavi and Waldeck [30]. The hydrodynamic friction is calculated
without the inclusion of any hydration layer. We have found that the
combined effect of dielectric and hydrodynamic friction gives an estimate close to the experimental result. This approach seems to provide
11
a microscopic basis for the standard hydrodynamic approach, where
a hydration layer is added to the protein in an ad hoc manner, to
calculate rotational friction.
The calculations adopted here are still not without limitations. The
continuum calculation of dielectric friction is dependent on the assumed cavity radius. Unfortunately, there is yet no microscopic basis
to assume certain value of the cavity radius for the calculation of dielectric friction. Moreover, the effect of increasing dielectric constant
of the solvent from the vicinity of the protein to the bulk is not taken
into account by Alavi and Waldeck [30]. Thus, we have attempted
to incorporate a multi shell model to incorporate multiple shells with
varying dielectric constants. The theory is described in the appendix
in detail. The drawback of incorporation of multiple shells in the continuum is that the frictional contributions from each of the shells add
up, thereby giving rise to an unphysical large result.
Similarly, the tri-axial method and bead modeling method suffer
from the lack of microscopic basis to determine the exact values of the
axes and the bead size, respectively.
A potentially powerful approach to the problem is the mode coupling theory [40], which uses the time correlation formalism to obtain
the memory kernel of the rotational friction. The total torque is separated into two parts – a short range part (which is called the bare
friction Γbare ) and a long range dipolar part. The advantage of mode
coupling theory is that it does not depend on any parameter. It uses a
time dependent effective potential field in terms of density distribution
and the direct correlation function given by [40],
Vef f (r, Ω, t) = −kB T
Z
′
′
′
′
′
′
dr dΩ c(r − r , Ω, Ω )∇Ωρ(r , Ω , t)
(8)
The torque density is then expressed as,
Nc (r, Ω, t) = n(r, Ω, t) −∇Ω Vef f (r, Ω, t)
(9)
where, n(r, Ω, t) is the number density of the tagged particle. The
rotational friction comes from the torque-torque correlation function.
12
The final expression of the single particle (Γs) and collective friction
(Γc ) are given by [41],
Γs (z) = Γbare +
Γc (z) = Γbare + A
Z
Z ∞
−zt ∞
dk
A
e
0
0
Z ∞
Z ∞
dk
e−zt
0
0
k2
k2
X
l1 l2 m
X
l1 l2 m
c2l1 l2 m (k)Fl2m (k, t)
(10)
Fls1m (k, t) c2l1 l2 m (k) Fl2 m (k, t)
(11)
. cl1 l2 m is the l1 l2m-th coefficient of the two particle
where, A =
direct correlation function between any two dipolar molecules. Fls1 l2 m
and Fl2 m (k, t) are the single particle and the collective orientational
correlation functions, respectively.
ρ
2 (2π)4
Eq. 10 and Eq. 11 are the standard mode coupling theory expressions for rotational friction. It has to be solved self consistently. In
the overdamped limit, the self dynamic structure factor is expressed
as,
kB T l(l + 1) −1
s
(12)
Flm(k, z) = z +
IΓs (z)
and the collective dynamic structure factor is given by,
c
Flm
= Flm (k) z +
kB T fllm(k)l(l + 1) kB T k 2fllm(k) −1
+
IΓc (z)
MΓT (z)
(13)
where fllm(k) = 1 − (−1)m(ρ/4π)cllm(k). I and M are the moment
of inertia and the mass of the dipolar molecule, respectively. ΓT (z) is
the frequency dependent translational friction.
The advantage of the mode coupling approach is that the once the
charge density of the protein molecules and the dipole density of the
water molecules surrounding the protein are defined, the rotational
friction can be obtained in terms of the direct correlation function and
the static and dynamic structure factors of the protein-water systems.
These are again related by Ornstein-Zernike equation [42].
13
The important aspect of this microscopic theory of dielectric friction is the hidden contribution of the translational modes. In the
hydration layer, the rotational friction is enhanced due to the slow
translational component. This effect of translation could not be approached through continuum calculation. Work in this direction is
under progress.
4
Appendix : Multiple shell model and the Drawback
Dielectric constant of water varies from the vicinity of the protein to
the bulk water value. To understand the effect of this varying dielectric
constant on the rotational dielectric friction of the protein, we have
performed the continuum calculation of rotational dielectric friction
using a multiple shell model.
Nee and Zwanzig derived the dielectric friction contribution of a
point dipole [27]. Alavi et al. [30] generalized it to obtain dielectric
friction of a molecule with arbitrary distribution of charges. Castner
et al. [43] generalized the point dipole approach to incorporate the
discrete shell model with varying dielectric constant. Here, we have
combined the approach of Castner et al. and Alavi et al. to obtain a
generalized arbitrary charge distribution model for multiple hydration
layers with varying dielectric constants around the protein. Figure 2
shows the general scheme of this work. The protein is in the innermost
cavity of radius a, where the water has a dielectric constant value of 4.
The dielectric constant of water in the successive layers is assumed to
increase up to the value of the bulk water, having a dielectric constant
of 78. The width of each shell is assumed to be d.
We first write down the electrostatic potential in two dimensions,
which could be generalized to three dimensions using principle of su-
14
perposition. The electrostatic potential Φj (r) can be written as,
Φj (r) = Φj (r, θ) =
∞
X
Blj
l=0
Pl (cos θ)
+ Ajl rl Pl (cos θ)
l+1
r
(14)
where, j denotes the number of concentric shells surrounding the protein. For n concentric shell, j can have a value from 0 to n + 1. j = 0
denotes no boundary. The boundary conditions are,
(i) Bl0 = qi ril , where qi and ri are the partial charge and the position
of the ith atom, respectively.
(ii) Φ → 0 as r → ∞
(iii)Φj (rj ) = Φj+1(rj+1), for j = 0, 1, 2...n.
′
′
′
(iv) ǫj Φj (rj ) = ǫj+1Φj+1(rj ), for j = 0, 1, 2..n, Φj (r) = ∂Φ∂rj (r) .
= 0,
(v ) An+1
l
After incorporating the boundary conditions in Eq. 14, we get,
Ajl
Blk
ǫk+1/ǫk − 1 =−
l
2l+1 ×
ǫk+1/ǫk + l+1
k=j rk
n
X
where,
Blj = qi ril
1
2l + 1 j j Πk=1
l+1
ǫk /ǫk−1 +
l
l+1
(15)
(16)
The reaction potential is given by,
Φj (r, θ, φ) =
N X
∞
X
i=1 l=0
A0l rl Pl (cos γi )
(17)
where,
Pl (cos γi) =
X
4π m=l
Ylm∗ (θi, φi)Ylm (θi, φi)
2l + 1 m=−l
(18)
After few steps of algebra, we obtain the frequency dependent dielectric friction given below,
ζlm (ω)
N X
N X
∞ X
l
X
2qiqj ri l rj l
×
=
a
a
j=1 i=1 l=1 m=1 a ω
(l − m)! m
Pl (cosθi )Plm (cosθj ) m cos(mφji ) ×
(l + m)!
15
n X
Im
a
s=0
Πsk=1 1 −
a 2l+1 ǫs+1,s(m ω) − 1 ×
l
+ sd
ǫs+1,s(m ω) + l+1
ǫk,k−1(m ω) − 1 l
ǫk,k−1(m ω) + l+1
(19)
where, ǫj,j−1 = ǫj /ǫj−1, for all values of j. ǫ0 is the dielectric constant
of the cavity.
Above is the general expression of multiple (n) shell model. To
write the final expression of dielectric friction for a two shell model,
we assume Debye relaxation for the frequency dependent dielectric
friction of two shells as given below,
ǫ1,0(mω) = 1 +
ǫ1,0 − 1
1 + i mω τD1
(20)
ǫ2,1 − 1
(21)
1 + i mω τD2
are the Debye relaxation time for the first and
ǫ2,1(mω) = 1 +
where, τD1 and τD2
second shell.
The expression of dielectric friction for a two-shell model is given
below,
ζDF
N X
N X
∞ X
l
X
l l
8 2l + 1 (l − m)!
rj
ri
×
=
qi qj
l + 1 (l + m)!
a
a
j=1 i=1 l=1 m=1 a
m2 Plm (cos θi ) Plm (cos θj ) cos(mφji ) ×
ǫ1,0 − 1
a 2l+1 ǫ2,1 − 1
τD1 +
τD2
(2ǫ1,0 + 1)2
a+d
(2ǫ2,1 + 1)2
ǫ2,1 − 1
a 2l+1 ǫ1,0 − 1
×
+2
a+d
(2ǫ1,0 + 1)2 (2ǫ2,1 + 1)2
(2ǫ1,0 + 1)τD2 + (2ǫ2,1 + 1)τD1 ,
16
(22)
The above expression has been numerically evaluated to find out
the effect of dielectric friction on protein due to varying dielectric
constant of water around the protein. The multiple shell model is
found to overestimate the dielectric friction, as is evident from the
above expression.
Acknowledgment The work is supported by DST, DBT and CSIR.
A.M. thanks CSIR for Senior Research Fellowship.
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20
Table 1
Table for the dielectric friction. The unit is 10−23 erg-sec.
Cavity radius is chosen such a way that the ratio of longest bond
vector (Rmax) of the protein to the chosen cavity radius (RC ) is 0.75.
X
Y
Z
av
Molecule RC (Å) ζDF
ζDF
ζDF
ζDF
6pti
29.50
17.8 13.2 18.1 16.0
1ig5
26.10
43.3 36.6 39.1 39.5
1ubq
34.30
18.1 18.3 21.8 19.3
351c
25.50
52.3 41.0 41.9 44.5
1pcs
27.20
90.5 51.3 66.1 65.7
1a1x
33.10
63.0 68.9 49.5 59.3
1gou
32.20
43.8 67.8 103.6 63.5
1aqp
35.30
44.5 71.1 132.1 68.0
1e5y
33.10
98.9 70.6 89.9 84.7
1bwi
35.70
78.3 60.5 108.1 77.8
1b8e
33.50 113.3 112.2 110.5 112.0
4ake
50.30
76.1 170.8 123.4 110.7
3rn3
35.00 118.8 89.0 56.8 80.5
1mbn
28.00 170.7 162.0 160.6 164.3
21
Table 2
Cavity size dependence of the dielectric friction. The unit is
10−23 erg-sec.
Molecule
6pti
1ig5
1ubq
351c
1pcs
1a1x
1gou
1aqp
1e5y
1bwi
1b8e
4ake
3rn3
1mbn
6lyz
0.75
ζDF
16.4
39.7
19.4
45.1
69.3
60.5
71.7
82.6
86.5
82.3
112.0
123.4
88.2
164.5
107.8
22
0.85
ζDF
25.7
61.3
30.3
69.3
111.0
96.4
114.6
132.3
136.4
128.9
174.1
211.7
138.1
263.1
172.7
Table 3
Table for the stick hydrodynamic friction using tri-axial ellipsoid. The unit is 10−23 erg-sec.
Molecule Rγ (Å)
6pti
11.34
1ig5
11.36
1ubq
11.73
351c
11.51
1pcs
12.38
1a1x
13.47
1gou
13.61
1aqp
14.45
1e5y
13.81
1bwi
13.94
1b8e
14.70
4ake
19.59
3rn3
14.31
1mbn
15.25
ζTAR
57.8
72.9
71.2
77.3
78.9
120.8
103.3
117.7
108.9
106.9
167.5
298.3
112.9
163.7
23
ζTBR
83.4
78.9
89.9
84.5
106.5
127.3
141.7
171.1
145.7
155.4
172.5
422.7
166.5
181.2
ζTCR
85.1
84.9
94.0
85.3
111.3
143.8
148.2
177.0
155.3
158.2
178.2
442.8
172.2
210.1
ζTavR
73.1
78.6
83.8
82.2
96.6
129.9
127.7
150.1
133.4
135.7
172.6
376.1
145.1
183.1
Table 4
Comparison between the total friction and the experimental
results. Results are given in the unit of 10−23 erg-sec. The
references to the experimental results of rotational diffusion of the
corresponding proteins are given in the Ref. [8] .
av
Protein
PDB id ζDF
ζTavR ζtotal ζexp
Bovine pancreatic trypsin inhibitor
6pti
16.0 73.1 89.1 96.8
Calbindin D9k, holo form
1ig5
39.5 78.6 118.1 125.0
Human ubiquitin
1ubq
19.3 83.8 103.1 118.9
Ferricytochrome c551
351c
44.5 82.2 126.7 130.1
Plastocyanin, Cu(II) form
1pcs
65.7 96.6 162.3 149.5
M T CP 1
Oncogenic protein p13
1a1x
59.3 129.9 189.2 241.9
Binase
1gou
63.5 127.7 191.2 191.3
Ribonuclease A
1aqp
68.0 150.1 218.1 186.1
Azurin, Cu(I) form
1e5y
84.7 133.4 218.1 190.4
Hen egg-white lysozyme
1bwi
77.8 135.7 213.5 203.6
Bovine -lactoglobulin, monomer
1b8e 112.0 172.6 284.6 270.6
Adenylate kinase, apo form
4ake 110.7 376.1 486.8 478.2
Bovine Ribonuclease A
3rn3
80.5 145.1 225.6 235.0
Sperm Whale Myoglobin
1mbn 164.3 183.1 347.4 246.3
24
500
300
ζexp (10
-23
Erg-Sec)
400
200
100
0
0
100
200
av
300
av
ζtotal = (ζDF + ζTR ) (10
-23
400
500
Erg-Sec)
Figure 1: The combined friction from hydrodynamic and dielectric is plotted against
the experimental results. The solid line shows the diagonal to guide the eye.
25
j=4
j=3
a+3d
j=1
a
j=2
a+d
a+2d
Bulk
Figure 2: schematic diagram of the Molecular cavity and the hydration shell constituted by the bound water molecules. The bulk water molecules are more randomly
oriented
26